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Solutions
59 Videos

Simplify.

`(7x^2 - 2x + 1)+(9x^2 -4x + 5)-(4x^2 + 6x - 7)`

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0.44mins

Q1a

Simplify.

`(7a^2 -4ab + 9b^2)-(-a^2+2ab + 6b^2)`

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0.37mins

Q1b

Determine two non-equivalent polynomials `f(x)`

and `g(x)`

, such that `f(0) =g(0)`

and `f(1) = g(1)`

.

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2.23mins

Q2

Ms. Frizzle has three daughters: Allison, Belle, and Claire. Today, January 1, their ages are, respectively,

```
\displaystyle
\begin{array}{cccccc}
& A(n) = -(n + 30)+(2n + 5) \\
& B(n) = (7 -n)-(32 -2n) \\
& C(n) = (n - 26)-(n + 4) + (n - 3) \\
\end{array}
```

All ages are expressed in years, and n represents Ms. Frizzle's age.

a) Are the daughter triples? Explain.

b) Are any of them twins? Explain.

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1.05mins

Q3ab

Ms. Frizzle has three daughters: Allison, Belle, and Claire. Today, January 1, their ages are, respectively,

```
\displaystyle
\begin{array}{cccccc}
& A(n) = -(n + 30)+(2n + 5) \\
& B(n) = (7 -n)-(32 -2n) \\
& C(n) = (n - 26)-(n + 4) + (n - 3) \\
\end{array}
```

All ages are expressed in years, and n represents Ms. Flangan's age.

(c) How old was Ms. Flanagan when Cassandra was born?

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0.33mins

Q3c

Expand and simplify.

`-3(7x -5)(4x - 7)`

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0.59mins

Q4a

Expand and simplify.

`-(y^2-4y + 7)(3y^2-5y-3)`

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1.45mins

Q4b

Expand and simplify.

`2(a +b)^3`

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1.21mins

Q4c

Expand and simplify.

`3(x^2 -2)^2(2x - 3)^2`

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2.38mins

Q4d

The volume of a cone is given by `\displaystyle V = \frac{1}{3}\pi r^2h`

. Determine the volume of the cone in simplified form if the radius is increased by `x`

and the height is increased by `2x`

.

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2.42mins

Q5

Simplify.

`(2x^4-3x^2 - 6)+(6x^4 -x^3+4x^2 + 5)`

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0.38mins

Q6a

Simplify.

`(x^2-4)(2x^2+ 5x - 2)`

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0.35mins

Q6b

Simplify.

`-7x(3x^3-7x + 2)-3x(2x^2-5x+ 6)`

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0.43mins

Q6c

Simplify.

`-2x^2(3x^3-7x+2)-x^3(5x^3+2x -8)`

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0.50mins

Q6d

Simplify.

`-2x[5x - (2x -7)] + 6x[3x-(1 + 2x)]`

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0.45mins

Q6e

Simplify.

`(x + 2)^2(x- 1)^2 - (x-4)^2(x + 4)^2`

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2.25mins

Q6f

Simplify.

`(x^2+5x - 3)^2`

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1.21mins

Q6g

Factor.

`12m^2n^3 + 18m^3n^2`

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0.30mins

Q7a

Factor.

`x^2-9x+20`

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0.25mins

Q7b

Factor.

`3x^2+24x + 45`

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0.30mins

Q7c

Factor.

`50x^2-72`

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0.34mins

Q7d

Factor.

`9x^2-6x + 1`

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0.11mins

Q7e

Factor.

`10a^2+a - 3`

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0.21mins

Q7f

Factor.

`2x^2y^4-6x^5y^3+8x^3y`

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0.33mins

Q8a

Factor.

`2x(x + 4)+3(x + 4)`

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0.12mins

Q8b

Factor.

`x^2 -3x-10 `

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0.12mins

Q8c

Factor.

`15x^2-53x +42`

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1.20mins

Q8d

Factor.

`a^4 -16`

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0.25mins

Q8e

Factor.

`(m - n)^2-(2m + 3n)^2`

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0.59mins

Q8f

Simplify. State any restrictions on the variables.

`\displaystyle \frac{10a^2b + 15bc^2}{-5b}`

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0.49mins

Q9a

Simplify. State any restrictions on the variables.

`\displaystyle \frac{130x^2y^3-20x^2z^2+50x^2}{10x^2}`

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0.37mins

Q9b

Simplify. State any restrictions on the variables.

`\displaystyle \frac{xy - xyz}{xy}`

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0.23mins

Q9c

Simplify. State any restrictions on the variables.

`\displaystyle \frac{16mnr-24mnp+40kmn}{8mn}`

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0.42mins

Q9d

Simplify. State any restrictions on the variables.

`\displaystyle 8xy^2 + 12x^2y - \frac{6x^3}{2xy}`

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1.19mins

Q10a

Simplify. State any restrictions on the variables.

`\displaystyle \frac{7a-14b}{2(a - 2b)}`

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0.26mins

Q10b

Simplify. State any restrictions on the variables.

`\displaystyle \frac{m + 3}{m^2+10m + 21}`

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0.26mins

Q10c

Simplify. State any restrictions on the variables.

`\displaystyle \frac{4x^2-4x -3}{4x^2-9}`

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0.42mins

Q10d

Simplify. State any restrictions on the variables.

`\displaystyle \frac{3x^2-21x}{7x^2-28x + 21}`

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0.38mins

Q10e

Simplify. State any restrictions on the variables.

`\displaystyle \frac{3x^2-2xy-y^2}{3x^2+4xy+y^2}`

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0.47mins

Q10f

Simplify. State any restrictions on the variables.

`\displaystyle \frac{6x}{8y}\times \frac{2y^2}{3x}`

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0.31mins

Q12a

Simplify. State any restrictions on the variables.

`\displaystyle \frac{10m^2}{3n}\times \frac{6mn}{20m^2}`

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0.26mins

Q12b

Simplify. State any restrictions on the variables.

`\displaystyle \frac{2ab}{5bc} \div\frac{6ac}{10b}`

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1.02mins

Q12c

Simplify. State any restrictions on the variables.

`\displaystyle \frac{5p}{8pq} \div \frac{3p}{12q}`

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0.43mins

Q12d

Simplify. State any restrictions on the variables.

`\displaystyle \frac{x^2}{2xy}\times \frac{x}{2y^2} \div \frac{(3x)^2}{xy^2}`

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0.54mins

Q13a

Simplify. State any restrictions on the variables.

`\displaystyle \frac{x^2-5x + 6}{x^2-1} \times \frac{x^2-4x -5}{x^2-4} \div \frac{x-5}{x^2+3x+2}`

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1.45mins

Q13b

Simplify. State any restrictions on the variables.

`\displaystyle \frac{1-x^2}{1+y} \times \frac{1-y^2}{x + x^2} \div \frac{y^3-y}{x^2}`

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2.00mins

Q13c

Simplify. State any restrictions on the variables.

`\displaystyle \frac{x^2-y^2}{4x^2-y^2}\times \frac{4x^2+8xy+3y^2}{x + y}\div \frac{2x + 3y}{2x -y}`

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1.47mins

Q13d

Simplify. State any restrictions on the variables.

`\displaystyle \frac{4}{5x}-\frac{2}{3x}`

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0.25mins

Q14a

Simplify. State any restrictions on the variables.

`\displaystyle \frac{5}{x + 1}- \frac{2}{x -1}`

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0.47mins

Q14b

Simplify. State any restrictions on the variables.

`\displaystyle \frac{1}{x^2+3x -4} + \frac{1}{x^2+x-12}`

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0.58mins

Q14c

Simplify. State any restrictions on the variables.

`\displaystyle \frac{1}{x^2-5x+6} - \frac{1}{x^2 - 9}`

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1.24mins

Q14d

Simplify and state any restrictions on the variables.

```
\displaystyle
\frac{1}{2x} - \frac{7}{3x^2}+ \frac{4}{x^3}
```

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1.11mins

Q15a

Simplify and state any restrictions on the variables.

```
\displaystyle
\frac{3x}{x + 2} + \frac{4x}{x -6}
```

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0.53mins

Q15b

Simplify and state any restrictions on the variables.

```
\displaystyle
\frac{6x}{x^2 - 5x + 6} - \frac{3x}{x^2 + x - 12}
```

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Q15c

Simplify and state any restrictions on the variables.

```
\displaystyle
\frac{2(x-2)^2}{x^2 + 6x + 5} \times \frac{3x + 15}{(2- x)^2}
```

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Q15d

Simplify and state any restrictions on the variables.

```
\displaystyle
\frac{(x - 2y)^2}{x^2 -y^2} \div \frac{(x- 2y)(x + 3y)}{(x + y)^2}
```

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2.02mins

Q15e

Simplify and state any restrictions on the variables.

```
\displaystyle
\frac{2b -5}{b^2 -2b -15} + \frac{3b}{b^2 + b - 30} \times \frac{b^2 + 8b + 12}{b + 3}
```

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2.02mins

Q15f

Fred’s final mark in an online course was determined entirely by two exams. The mid-term exam was out of `x`

marks and was worth 25% of his final mark. The final exam was out of `2x`

marks and was worth 75% of his final mark. Fred scored 40 marks on the first exam and 60 marks on the second exam. Determine the value of `x`

if Fred earned a final mark of 50% in the course.

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3.27mins

Q16

Sam plays a game in which he selects three different numbers from `1`

to `n`

(`n > 3`

). After he selects his numbers, four different winning numbers from `1`

to `n`

are chosen, one at a time. Sam wins if all three of his numbers are among the four winning numbers.

The first number chosen is one of Sam's! His probability of winning is now given by

```
\displaystyle
P(n) = \frac{24}{n^3 - 3n^2 + 2n} \div \frac{3}{n}
```

a) Simplify P(n) and state restrictions on `n`

.

b) What would Sam's probability of winning be if

- i) n = 5?
- ii) n = 4?

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3.09mins

Q17

Product of Rational Expressions
9 Videos

**Definition** Rational number is any number that can be expressed as `\frac{a}{b}`

where `a, b`

, are integers.

*ex* `\dfrac{x + 1}{x - 1}`

, `\dfrac{x^2 - 2x - 3}{(x + 2)(x - 5)}`

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3.50mins

Introduction to Rational Expressions

*ex* Simplify & state restrictions

`\dfrac{AB}{BC} \times \dfrac{C}{D} = \dfrac{A}{D}, B \neq 0, C \neq 0`

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2.37mins

Simplifying Rational Expressions

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3.09mins

Simplying Rational Expressions and Stating Restricitons

*ex* Simplify and state the restrictions.

`\dfrac{(2x -1)(x + 5)}{(x -1)(x + 5)}`

`= \dfrac{2x - 1}{x - 1}, x \neq 1, -5`

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0.40mins

Simplifying Rational Expressions and its restrictions

*ex* Simplify and state the restrictions.

`\dfrac{2x^2 -x -3}{x^2 + 2x +1}`

`= \dfrac{2x - 3}{x + 1}, x \neq -1`

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1.04mins

Simplifying Rational Expressions by Factoring and its Restrictions

*ex* Simplify and state the restrictions.

`\dfrac{x^2 -4}{(x + 6)^2} \cdot \frac{x^2 + 9x + 18}{2(2 -x)}`

`= \dfrac{(x + 2)(x + 3)}{-2(x + 6)}, x \neq -6, 2`

, x ∈ `\mathbb{R}`

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1.40mins

Multiplying Rational Expressions

```
\displaystyle
\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C} = \frac{AB}{BC}
```

where
```
\displaystyle
B, C, D =\neq 0
```

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2.50mins

Restriction on Division of Rational Expressions

*ex* Simplify and state the restrictions.

```
\displaystyle
\frac{(x + 1)(x-1)}{x+ 5} \div \frac{x -1}{(x + 5)(x -3)}
```

Therefore, `(x + 1)(x - 3), x \neq -5, 1, 3`

.

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2.17mins

Division of Rational Expression Example

Sum of Rational Expressions
6 Videos

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A overview Introduction

`\dfrac{A}{B} + \dfrac{C}{D} = \dfrac{AD + BC}{BD}`

*ex* Combine and simplify

`\dfrac{1}{x -1} + \dfrac{1}{2x + 1}`

`= \dfrac{2x + 1 + x - 1}{(x - 1)(2x + 1)}`

`= \dfrac{3x}{(x - 1)(2x + 1)}`

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2.17mins

1 Introduction to Sum of Rational Expressions

`\dfrac{A}{BC} + \dfrac{D}{BE} = \dfrac{AE + CD}{BCE}`

*ex* Combine and simplify.

`\dfrac{1}{x^2 - 1} + \dfrac{1}{x^2 - x -2}`

`= \dfrac{2x - 3}{(x - 1)(x + 1)(x - 2)}`

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4.02mins

2 Adding Rational Expression with more than one factors in the denominator

*ex* Subtract and simplify.

`\dfrac{1}{(x -1)(x + 2)^2} - \dfrac{1}{(x -2)(x + 2)(x - 1)}`

`= \dfrac{-4}{(x - 1)(x + 2)^2(x - 2)}`

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1.39mins

3 Subtraction of Rational Expressions

*ex* Add and simplify.

`\dfrac{2}{(x^2 -4)(x^2-4x +3)} + \frac{1}{(x^2 -x -2)(x^2 + 4x + 4)}`

`= \dfrac{3x^2 + 2x + 7}{(x-2)(x + 2)^2(x-1)(x + 1)(x-3)}`

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2.35mins

4 Addition of Two Rational Terms

*ex* Find the sum.

`\dfrac{1}{x - 2} + \dfrac{1}{(x - 2)^2} + \dfrac{1}{(x - 2)^3}`

`= \dfrac{x^2 - 3x + 3}{(x - 2)^3}`

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2.27mins

5 Adding Three Rational Terms